18.6 Internally Represented Cyclotomics

The implementation of an internally represented cyclotomic is based on a list of length equal to its conductor. This means that the internal representation of a cyclotomic does not refer to the smallest number field but the smallest cyclotomic field containing it. The reason for this is the wish to reflect the natural embedding of two cyclotomic fields into a larger one that contains both. With such embeddings, it is easy to construct the sum or the product of two arbitrary cyclotomics as an element of a cyclotomic field.

The disadvantage of this approach is that the arithmetical operations are quite expensive, so the use of internally represented cyclotomics is not recommended for doing arithmetics over number fields, such as calculations with matrices of cyclotomics. But internally represented cyclotomics are good enough for dealing with irrationalities in character tables (see chapter Character Tables).

For the representation of cyclotomics one has to recall that the n-th cyclotomic field Q(e_n) is a vector space of dimension j(n) over the rationals where j denotes Euler's phi-function (see Phi).

A special integral basis of cyclotomic fields is chosen that allows one to easily convert arbitrary sums of roots of unity into the basis, as well as to convert a cyclotomic represented w.r.t. the basis into the smallest possible cyclotomic field. This basis is accessible in GAP, see Integral Bases for Abelian Number Fields for more information and references.

Note that the set of all n-th roots of unity is linearly dependent for n > 1, so multiplication is not the multiplication of the group ring Qáe_n ñ; given a Q-basis of Q(e_n) the result of the multiplication (computed as multiplication of polynomials in e_n, using (e_n)ⁿ = 1) will be converted to the basis.

gap> E(5) * E(5)^2; ( E(5) + E(5)^4 ) * E(5)^2;
E(5)^3
E(5)+E(5)^3
gap> ( E(5) + E(5)^4 ) * E(5);
-E(5)-E(5)^3-E(5)^4

An internally represented cyclotomic is always represented in the smallest cyclotomic field it is contained in. The internal coefficients list coincides with the external representation returned by ExtRepOfObj.

Since the conductor of internally represented cyclotomics must be in the category IsSmallIntRep, the biggest possible (though not very useful) conductor is 65535. So the maximal cyclotomic field implemented in GAP is not really the field Q^ab.

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